Transcription of 量子理工学専攻量子力学 II - Nihon University
1 II 2020 ( : 2020 9 11 ) v vii ix 1 1 .. Hermite ..4 ..9 2 3 SO(3) Lie so(3).. 3 .. Lie so(3) .. Wigner D .. clebsch -Gordan ..31 ..36 3 Coulomb .. Coulomb .. Schr dinger ..52 ..53 4 55 ..57 5 .. S .. M ller S .. Schr dinger S Dyson .. Heisenberg S .. Lippmann-Schwinger .. S ..78iv ..81 ..82 6 .. 1 .. 2 .. Fermi ..88 ..91 A ..95v I ( ) 6 : 1 1 1 Schr dinger 2 Wigner D clebsch -Gordan 3 Coulomb Coulomb ( ) Coulomb 4 Pauli 5 S Lippmann-Schwinger 6 ( ) ( ) vii 2 :[1]J.
2 J. Sakurai and J. Napolitano,Modern Quantum Mechanics, 2nd ed. (Cambridge University Press, 2017)[2]S. Weinberg,Lectures on Quantum Mechanics, 2nd ed. (Cambridge University Press, 2015) [1] 2 John R. Taylor :[3]J. R. Taylor,Scattering Theory: The Quantum Theory of Nonrelativistic Collisions(Dover Publications, 2006) 1972 ix cgs-Gauss MKSA MKSA cgs-Gauss E B j cgs-Gauss Maxwell : E= 4 B 1c@tE=4 cj B= 0 E+1c@tB= 0 2 q1 q2 Coulomb cgs-Gauss Coulomb F :F=q1q2r2rr Coulomb V :V=q1q2r d(x) = ddp(2 h)dei hp x1 1 1 Hilbert H Hilbert Hilbert *1 H H H H 3 H H H S ( ) 1 ( ) *2 3 5 2 m !
3 1 H :H=p22m+m!22x2( ) x p :[x;p] =i h;[x;x] = 0;[p;p] = 0( ) HjE =EjE 2 :1 :H=p22m+m!22x2= h!(m!2 hx2+12 hm!p2)= h![( m!2 hx ip2 hm!p)( m!2 hx+ip2 hm!p) i2 h(xp px)]= h!(aya+12)( )*1 Hilbert (unit ray) (density matrix) (self-adjoint operator) Stone *2 H (domain) H 2 1 1 [x;p] =xp px=i h a ay :a:= m!2 hx+ip2 hm!p( )ay:= m!2 hx ip2 hm!p( )ay a 2 H a ay a ay a ay :[a;a] = 0and[ay;ay] = 0( ) a ay ( ) :[a;ay] =[ m!]
4 2 hx+ip2 hm!p; m!2 hx ip2 hm!p]=m!2 h[x;x] i2 h[x;p] +i2 h[p;x] +12 hm![p;p]=m!2 h 0 i2 h i h+i2 h ( i h) +12 hm! 0= 1( ) 2 3 ( ) :( ) [A; B+ C] = [A;B] + [A;C]( )[ A+ B;C] = [A;C] + [A;C]( )( ) [A;B] = [B;A]( ) A B C c a ay H a ay :[H;a] =[ h!(aya+12);a]= h![aya;a]= h!([ay;a]a+ay[a;a])= h!( 1 a+ay 0)= h!a( )[H;ay] =[ h!(aya+12);ay]= h![aya;ay]= h!([ay;ay]a+ay[a;ay])= h!(0 a+ay 1)= + h!ay( ) 3 8 Leibniz :*3(Leibniz ) [A;BC] = [A;B]C+B[A;C]( )[AB;C] = [A;C]B+A[B;C]( ) :HjE =EjE ( )*3 ( ) ( ) Leibniz ( ) Leibniz ddx(fg) = (ddxf)g+f(ddxg) ( ( ) A ddx B C f g ) ( ) [aya;a] Leibniz [aya;a] =ayaa aaya 3 2 :1 ay a h!
5 ( ) jE a ay ajE ayjE H :HajE = ([H;a] +aH)jE = ( h!a+aE)jE = (E h!)ajE ( )HayjE =([H;ay] +ayH)jE =(+ h!ay+ayE)jE = (E+ h!)ayjE ( ) 2 5 H E ( ) ajE H E h! ( ) ayjE H E+ h! a h! ay h! ay a 2 jE jE a n n h! anjE /jE n h! ay n n h! (ay)njE / jE+n h! : ay ! ajE n h! ay ! a ay ! ajE h! ay ! ajE ay ! ajE+ h! ay ! a ay ! ajE+n h! ay ! a ( )2 ajE 0 E 12 h! E=12 h! ajE = 0 ajE 2 : ajE 2= EjayajE = Ej(1 h!H 12)jE =(E h! 12) EjE =E h! 12 0( ) jE jE 2= EjE = 1 ajE 0 E :E 12 h!( ) ( ) E h!
6 12= 0 ajE :*4E=12 h!,ajE = 0( ) E 1 2 jE a E=12 h! ( ) jE0 ajE0 = 0 :0 ajE0 ay ! ajE0+ h! ay ! a ay ! ajE h! ay ! ajE ay ! ajE+ h! ay ! a ( ) ( ) ajE0 = 0 E0 E0=12 h! E0 E1= (1 +12) h! E2= (2 +12) h! ( ) *4 ( ) 4 1 1 ( ) :En=(n+12) h!; n2f0;1;2; g( ) jEn jEn /(ay)njE0 :jEn =cnayjEn 1 ( ) cn jEn jEn 1 ( ) 2 :1 = EnjEn =jcnj2 En 1jaayjEn 1 =jcnj2 En 1j([a;ay] +aya)jEn 1 =jcnj2 En 1j(1 +H h! 12)jEn 1 =jcnj2(1 +En 1 h! 12) En 1jEn 1 =jcnj2(1 +n 1 +12 12)=jcnj2n( ) jcnj2= 1/n cn cn= 1/pn jEn =1pnayjEn 1 jEn :jEn =1pnayjEn 1 =1pnay(1pn 1ayjEn 2 )=1 n(n 1)(ay)2jEn 2 = =1 n(n 1) 1(ay)njE0 =1pn!
7 (ay)njE0 ( ) jE0 ajE0 = 0 jE0 jEn ( ) jEn 3 :1 ajE0 = 0 2 jE0 = 1 jE0 3 jE0 ay n jEn =1pn!(ay)njE0 Hermite x p Hilbert [x;p] =i h Hilbert R 2 *5 x p *5 R 2 L2(R) L2(R) ( ) Hilbert Hermite 5 ( ) :*6x!x( )p! i hddx( ) ( ) ( ) 1 :a! m!2 hx+ip2 hm!( i hddx)= + h2m!(ddx+m! hx)( )ay! m!2 hx ip2 hm!( i hddx)= h2m!(ddx m! hx)( ) W(x) *7ddx+W (x) =e W(x)ddxeW(x)( ) ( ) ( ) :a!+ h2m!e m!2 hx2ddxe+m!2 hx2( )ay! h2m!e+m!2 hx2ddxe m!2 hx2( ) j j :j ! (x)( ) j !
8 1 1dx (x) (x)( ) ( Hilbert 2 ) 3 ( ) jEn n(x) 1 ajE0 = 0 ( ) ajE0 = 0 0(x) 1 :ajE0 = 0! h2m!(ddx+m! hx) 0(x) = 0( ) : 0(x) =Ne m!2 hx2( ) N ( ) 2 jE0 = 1 jE0 N ( ) jE0 2= E0jE0 = 1 0(x) : jE0 2= 1! 1 1dxj 0(x)j2= 1( )*6 x p (x) :(x )(x) :=x (x)&(p )(x) := i hd dx(x) x p ( ; ) = 1 1dx (x) (x) ( ;x ) = (x ; ) ( ;p ) = (p ; ) (x) ([x;p] )(x) = ((xp px) )(x) = i hxd dx(x) +i hddx(x (x)) =i h (x) *7 (x) :e W(x)ddx(eW(x) (x))=e W(x)(W (x)eW(x) (x) +eW(x) (x))=(ddx+W (x)) (x) (x) e W(x)ddxeW(x)=ddx+W (x) 6 1 1 ( ) : ( ) =jNj2 1 1dxe m!
9 Hx2=jNj2 hm!( ) 2 Gauss 1 1dxe Ax2= A jNj2= m! h N N :N=(m! h)1/4( ) 3 jE0 ay n jEn =1pn!(ay)njE0 n(x) ( ) 1pn!(ay)njE0 0(x) n :jEn =1pn!(ay)njE0 ! n(x) =1pn!( 1)n( h2m!)n/2em!2 hx2ddxe m!2 hx2 em!2 hx2ddx(e m!2 hx2 0(x))=1pn!( 1)n( h2m!)n/2em!2 hx2dndxn(e m!2 hx2 0(x))=N 1pn!( 1)n( h2m!)n/2em!2 hx2dndxne m! hx2=Ne m!2 hx2 1pn!( 1)n( h2m!)n/2em! hx2dndxne m! hx2( ) 3 ( ) ( ) Ne m!2 hx2 0(x) = (2m! h)1/2x ddx=d dxdd = (2m! h)1/2dd dndxn= (2m! h)n/2dnd n ( ) ( h2m!)n/2em! hx2dndxne m! hx2 e12 2dnd ne 12 2 n n(x) ( ) ( ) : n(x) = 0(x)Hn( 2m! hx)( ) Hn :Hn( ) :=1pn!( 1)ne12 2dnd ne 12 2( ) Hn( ) n ( ) n Hermite (Hermite polynomial) *8 1 ( ) *9 Hermite ( ) n Hermite 1 1 *8 Hermite III (p.)
10 91) :Hn(x) = ( 1)ne12x2dndxne 12x2 ( ) 1/pn! 1 n(x) ( ) ( n ) Hermite ( ) 1 Schr dinger ( ) (n ) *9 1 1 Schr dinger Hermite 7 0 RezImz C(a) C 0 RezImzC (b) C : ( ) C C ( ) Hermite ( ) n dnd ne 12 2 n Cauchy :f(n)( ) =n!ICdz2 if(z)(z )n+1( ) C z= Cauchy dnd ne 12 2 :dnd ne 12 2=n!ICdz2 ie 12z2(z )n+1=n!IC dz2 ie 12(z+ )2zn+1=n!e 12 2IC dz2 ie 12z2 zzn+1=n!e 12 21 =01 m=0( 1) +m m2 !m!IC dz2 iz2 +m n 1=n!e 12 2[n2] =0( 1)n n 2 2 !(n 2 )!( ) 1 C z 2 z!z+ z= 0 C ( ) 3 e 12(z+ )2=e 12 2e 12z2 z 4 e 12z2 z=e 12z2e z z= 0 Taylor :e 12z2 z=e 12z2e z=[1 =01 !]
